Unformatted text preview: Problem Set 5 Chem 3900: Physical Chemistry II
Spring 2011 Due: Class, Friday, Mar 4th Please make sure to write your name and the recitation you’ll attend. When you print out your work, try to minimize the number of pages. You can do this by printing multiple pages per sheet and/or by using a duplex printer. Problem 1. In addition to his celebrated theories of special and general relativity, Albert Einstein made many other contributions to chemistry and physics. One of these is his theory of the heat capacity of crystalline solids, described in Example 17‐3 and Exercise 17‐20 of McQ&S. In the Einstein model of diamond, the vibrational motion of each carbon atom along x, y, and z directions is represented as a harmonic oscillator. The energy level of the harmonic oscillator was discussed in Chem 3890 (It is also shown in Figure 5.7 of McQ&S). Since each direction in space is independent and if we assume that the atoms are non‐interacting (except for providing a static harmonic potential), we can treat the problem as collection of 3N independent 1D oscillators. This problem concerns a collection of 1D harmonic oscillators at temperature T. The different oscillators occupy different quantum states, labeled by the quantum number v. The average value of this quantum number, <v >, is given by v vPv 0 with Pv the Boltzmann probability that the vibration has quantum number v. i) Show that ii) The vibrational frequency of diamond is = 2.75 ä 1013 s‐1. Calculate the corresponding vibrational temperature. / iii) Use your answer to (i) to determine the temperature at which each vibration in diamond is, on average, in its first excited state. How does this compare to (ii)? iv) Plot < v > as a function of T from 0 K to 2000K. Compare your result with Figure 17.4 of McQ&S. Can you use your plot to explain the temperature dependence of molar heat capacity of diamond? 1 Problem 2.
A class with 150 students (not Chem 3900!) is graded on a curve with a median
grade of B. The number of students receiving each type of grade in this case
would be, grade (number of students), A+(6), A(15), A−(22), B+(24), B(23),
B−(21), C+(15), C(12), C−(8), D (4). i) Suppose that instead of assigning grades based on performance, the instructor (not me!) decides to assign grades randomly. How many ways can the instructor assign grades, consistent with this curve? Your answer should be expressed in scientific notation. Following in the footsteps of Boltzmann, assign a dimensionless “grade entropy” to this distribution of grades, and calculate this entropy ii) iii) Now consider a different grading distribution in which the same number of students is assigned to each of the 10 grade categories listed above. Calculate the “grade entropy” for this distribution. iv) Compare your answers to (ii) and (iii). Explain, in words, why the grade entropy is larger for one distribution of grades than for the other. Without doing any calculations, describe in words what constitutes a “high‐entropy” grade distribution. For what distribution would the grade entropy equal zero? Problem 3.
To get a feel for the connection between the partition function q, pj, and the entropy, consider a model system with only 3 quantum levels: 1 = 0, 2 = f , and 3 = . We take 0 ≤ f ≤ 1. Plot q as a function of = 1/(kBT) in the ranges 0 § § 10 and 0 § § 100, where is measured in units of 1/. Use two values of f: 0.02 and 0.5. i) ii) Plot the probabilities of the three levels over the same temperature range for each f. iii) Now plot the dimensionless entropy s = S/kB over the same temperature range for each f. iv) Explain the form of your plots, in particular, the limiting behavior for → 0 and for large. What values do you expect for q and s as → ∞? 2 Problem 4. i)
vii) 1.00 mole of an ideal gas of diatomic molecules ( CV = (5/2)R) starts at T = 350 K and p = 1 bar. Calculate ΔS for each of the following processes. The gas is compressed isothermally and reversibly to 1/3 of its initial volume. The gas is subjected to a fixed external pressure of 5.0 bar and is compressed isothermally to 1/3 of its initial volume. The gas expands isothermally against no external pressure to twice of its initial volume The gas pressure is fixed at 1 bar, and the gas is cooled to 280 K. The gas volume is fixed at its initial value, and the gas is cooled to 280 K. The gas is compressed reversibly and adiabatically to 1/2 of its initial volume. The gas is subjected to a fixed external pressure of 5.0 bar and is compressed adiabatically to 1/2 of its initial volume. Problem 5. 1.00 mole of an ideal gas of nonlinear molecules ( CV = (7/2)R) starts at T = 400 K and p = 1 bar, is subjected to the following three‐step process: A. The gas expands isothermally and reversibly until the pressure has dropped to 0.8 bar. B. At a fixed external pressure of 0.8 bar, the gas is connected to a thermostat at 280 K, and the gas is cooled until its volume reaches its initial value at the start of step A. C. The gas is now placed in contact with a thermostat at 400 K, and is heated at fixed volume to 400 K. i)
Sketch this process on a pV plot. ii)
Determine q and w for each step. iii)
This cycle represents an engine. Why? iv)
Calculate the efficiency of this engine. Does your answer satisfy the Second Law of Thermodynamics? Explain. 3 ...
View Full Document